Sure, let's tackle these ratio problems step by step.
### Part (a)
Given ratios:
- [tex]\(A : B = 3 : 5\)[/tex]
- [tex]\(A : C = 6 : 7\)[/tex]
#### Step 1: Find a common base for [tex]\(A\)[/tex]
1. From [tex]\(A : B = 3 : 5\)[/tex], we can write [tex]\(A = 3k_1\)[/tex] and [tex]\(B = 5k_1\)[/tex].
2. From [tex]\(A : C = 6 : 7\)[/tex], we can write [tex]\(A = 6k_2\)[/tex] and [tex]\(C = 7k_2\)[/tex].
#### Step 2: Equalize the [tex]\(A\)[/tex] terms
To find the least common multiple (LCM) of the coefficients of [tex]\(A\)[/tex] in both ratios:
- The coefficients are 3 and 6.
- The LCM(3, 6) is 6.
#### Step 3: Adjust the terms
Adjust each ratio so that the [tex]\(A\)[/tex] term is the LCM:
1. [tex]\(A = 6\)[/tex]
- From [tex]\(A = 3k_1\)[/tex]: [tex]\(k_1 = 2\)[/tex]. So, [tex]\(B = 5 \times 2 = 10\)[/tex].
- From [tex]\(A = 6k_2\)[/tex]: [tex]\(k_2 = 1\)[/tex]. So, [tex]\(C = 7 \times 1 = 7\)[/tex].
Thus, the ratio [tex]\(A : B : C\)[/tex] is [tex]\(6 : 10 : 7\)[/tex].
### Part (b)
Given ratios:
- [tex]\(B : C = \frac{1}{2} : \frac{1}{6}\)[/tex]
- [tex]\(A : B = \frac{1}{3} : \frac{1}{5}\)[/tex]
#### Step 1: Simplify the ratios
1. Simplify [tex]\(B : C = \frac{1}{2} : \frac{1}{6}\)[/tex]:
- Convert to integers by multiplying by 6 (common denominator): [tex]\(B : C = 3 : 1 \)[/tex].
2. Simplify [tex]\(A : B = \frac{1}{3} : \frac{1}{5}\)[/tex]:
- Convert to integers by multiplying by 15 (common denominator): [tex]\(A : B = 5 : 3\)[/tex].
#### Step 2: Combine the ratios
We have:
- [tex]\(A : B = 5 : 3\)[/tex]
- [tex]\(B : C = 3 : 1\)[/tex]
#### Step 3: Equalize the [tex]\(B\)[/tex] terms
Since [tex]\(B\)[/tex] is already the same in both ratios, we can combine them directly.
1. From [tex]\(A : B\)[/tex], we have [tex]\(A = 5k\)[/tex] and [tex]\(B = 3k\)[/tex].
2. From [tex]\(B : C\)[/tex], we have [tex]\(B = 3m\)[/tex] and [tex]\(C = m\)[/tex].
Set the [tex]\(B\)[/tex] terms equal:
- [tex]\(3k = 3m\)[/tex]
- [tex]\(k = m\)[/tex]
Thus, [tex]\(A = 5k\)[/tex], [tex]\(B = 3k\)[/tex], and [tex]\(C = k\)[/tex].
Therefore, the ratio [tex]\(A : B : C = 5k : 3k : k = 1.2 : 3 : 1\)[/tex].
### Final Ratios:
- Part (a): [tex]\(6 : 10 : 7\)[/tex]
- Part (b): [tex]\(1.2 : 3 : 1\)[/tex]
